Start by taking out all the old grout.
You need to make sure the floor is stable, the grout is probably cracking because of a poor install, or poor subfloor. There may not be much you can do about that.
For your situation I'd recommend an epoxy grout, it will be a tiny bit flexible, and much more resilient than a cement based grout. But be careful when installing it, it sets up quick and is a pain to clean up if you don't follow the instructions and watch your timing.
Stay away from any premixed grouts if you want durability.
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